Answer
Assuming that the sound waves from each individual cheering are coherent and that they all arrive at your location in phase, we can use the inverse square law to relate the number of people cheering to the sound intensity level at your location:
IL = 10log(I/Io) = 10log((P/(4pir^2))/Io)
where IL is the sound intensity level, I is the sound intensity, Io is the reference intensity (1 x 10^-12 W/m^2), P is the total sound power emitted by all the individuals cheering, r is the distance from the center of the circle of people to your location, and pi is the mathematical constant pi.
We can rearrange this equation to solve for P:
P = 4pir^2*10^(IL/10)*Io
We know that the sound intensity level at your location is 87 dB, which is 7 dB higher than the sound intensity level that would be produced by a single individual cheering. Therefore, we can conclude that the number of people cheering is:
P = 4pir^2*10^(7/10)*Io
Let's assume that the radius of the circle of people is 10 meters (this is just an example, the actual value is not given in the problem). Then, we can calculate the number of people as:
P = 4pi(10^2)10^(7/10)Io = 4pi10,000*10^(7/10)*Io
P = 4pi10,00010^(7/10)(1 x 10^-12) = 1.26 x 10^-6
Therefore, the number of people cheering for you is approximately:
P = 1.26 x 10^-6 people
Since this is not a meaningful result, it is likely that our assumption of a circle with a radius of 10 meters was incorrect. However, we can use this equation to calculate the number of people cheering if we know the actual distance from the center of the circle to your location.
Work Step by Step
intensity
I = $ \frac{P}{4 \pi r} $
